Statistical decision problem

In mathematical statistics, a statistical decision problem is a triple made up of a statistical model , a decision space and a loss function . Statistical decision problems form the framework for finding optimal decision functions and are therefore a general superstructure for determining good range estimates , point estimates and the design of statistical tests .

definition

A triple is called a statistical decision problem if ${\displaystyle ({\mathcal {E}},\Omega ,L)}$

• ${\displaystyle {\mathcal {E}}}$is a statistical model , i.e. for a basic set , a σ-algebra on this basic set and a family of probability measures . ${\displaystyle {\mathcal {E}}=(X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$
• ${\displaystyle \Omega }$is the basic set of a decision space , i.e. a measurement space whose σ-algebra contains all point sets. ${\displaystyle \Sigma }$
• ${\displaystyle L:\Theta \times \Omega \to [0,+\infty ]}$is a loss function .

example

Consider the 100-fold product model as a statistical model, which models the repeated coin toss

${\displaystyle {\mathcal {E}}=(\{0,1\}^{100},{\mathcal {P}}(\{0,1\})^{\otimes 100},\operatorname {Ber} _{\vartheta }^{\otimes 100})}$,

The decision space is the measurement space that contains the parameter of the Bernoulli distribution to be estimated and, for example, the Gaussian loss as a loss function ${\displaystyle (\Omega ,\Sigma )=([0,1],{\mathcal {B}}([0,1]))}$ ${\displaystyle \operatorname {Ber} _{\vartheta }}$

${\displaystyle L(\vartheta ,\omega )=(\omega -\vartheta )^{2}}$,

so is a statistical decision problem. ${\displaystyle ({\mathcal {E}},\Omega ,L)}$

Statistical decision problems specially tailored to the estimation are also called estimation problems .

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .